\chapter*{Preface}

This is a course in mathematical proof. 
It is for math majors who are US sophomores, although since
it requires only high school mathematics
it can be used with first year students.



\medskip
\noindent\textsc{Approach.}
This course is inquiry-based (sometimes called Moore method 
or discovery method).
This text is a sequence of exercises,
along with definitions and a few remarks.
The students work through the material together by
proving statements or sometimes by providing examples or counterexamples.
This makes each student grapple directly with the 
mathematics\Dash the instructor only 
lightly guides, while the students pledge not to use outside sources\Dash
talking out misunderstandings, 
sometimes stumbling in the dark, and sometimes
having beautiful flashes of insight.
For these students, with this material,
this is the best way to develop mathematical maturity.
Besides, it is a lot of fun.


\medskip
\noindent\textsc{Topics.}
We start with elementary number theory, not logic and sets, 
for the same reason
that the baseball team's annual practice starts with tossing the ball and 
not with reading the rulebook.
Math majors take readily to proving things about
divisibility and primes, 
whereas a month of preliminary material can be less of a lure.

But the background is good stuff also and 
students are on board once they see where it is going.
In the second and third chapters we do
sets, functions, and relations, now leveraging the
intellectual habits that we've established at the start.



\medskip
\noindent\textsc{Exercises.}
The exercises are arranged so that, as much as the material allows,
their difficulty level is roughly constant.
However note that this level rises gradually  
through the book.

Some exercises have multiple items; these come in two types.
If the items are labeled \textsc{A}, \textsc{B}, etc., 
then each one is hard enough to be a separate assignment.
If the labels are (i), (ii), etc., then they together make
a single assignment.
(In my course I have students put proposed solutions on the board
for the group to discuss.
If the items are labelled alphabetically then I ask a different student
to do each, while for the others I ask one student to do them all.)



\medskip
\noindent\textsc{License.}
This material is Free; see \url{http://joshua.smcvt.edu/proofs}.
This includes the \LaTeX{} source so that an instructor can tailor 
the work to their students.  


\vspace{\fill}
\noindent\parbox{.95\textwidth}{\raggedright\textit{At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: "Mr Smith, please prove Theorem 1. Oh, you can't? Very well, Mr Jones, you? No? Mr Robinson? No? Well, let's skip Theorem 1 and come back to it later. How about Theorem 2, Mr Smith?" Someone almost always could do something. If not, class dismissed. It didn't take the class long to discover that Moore really meant it, and presently the students would be proving theorems and watching the proofs of others with the eyes of eagles.}\hspace{1.5em}---Paul Halmos}

\vspace{.2in}
\noindent\parbox{.95\textwidth}{\textit{It's a kind of art that may change lives.}\hspace{1.5em}---Peter Schjeldahl}
 
\vspace*{.4in}
\begin{flushright}
  \begin{tabular}{@{}l@{}}
  Jim Hef{}feron  \\
  Saint Michael's College  \\
  Colchester, Vermont USA \\
  2013-Spring
  \end{tabular}
\end{flushright}
